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Full text of "Mathematical And Physical Papers - Iii"

ON THE MOTION"  OF  PENDULUMS.                            69
The first part of this expression is evidently twice the work, during the time Dt, of the external forces which act all over the mass. The second part becomes after integration by parts
- Wtff(uPt + vT3 + wT^) dy dz - WtJf(vPz + wT^ + uTs) dz dx
- Wtff (wP3 + uT^ + vTJ dx dy
dy   2    dz    3    \dz    dyj   *    \dx    dzj   2
du , dv\ m} j   j   j -r + -r)Ts[d% dydz. dy    dx)   8j        *
The double integrals in this expression are to be extended over the whole surface S. If dS be an element of this surface, I ', m', n' the direction-cosines of the normal drawn outwards at dS, we may write VdS, m'dS, n'dS for dydz, dzdx, docdy. The second part of DV thus becomes
- Wtff [u (I'P^ 4- m'T9 + n'T^ + v (m'P, + n'T^ + VTj
The coefiScients of u, v, w in this expression are the resolved parts, in the direction of a?, y, z, of the pressure on a plane in the direction of the elementary surface dS, whence it appears that the expression itself denotes twice the work of the pressures applied to the surface of the portion of fluid that we are considering.
On substituting for PI} &c. their values given by the equations (133), (134), we get for the last part of DV
dv    dw\
- wt       2    \ + 2 d\ + 2      - 1    +
^                                                     *
+ 2      - 1    +   4
\dy)        \dz)     * \dx    dy    dz J
dv     dw\*     fdw    du\2     fdu    dv\^\   1    1   ,
T" + T"    + ( j- H- T~    + ( -j- + ;j- H ^ ^ ^-s-flk     dy/      \d^     d^/      Vdy    cte/ j        ^
In this expression p denotes, in the case of an elastic fluid, the pressure statically corresponding to the density which actually exists about the point whose co-ordinates are x, y, z, and the part of the expression which contains p denotes twice the work converted into vis viva in consequence of internal expansions, and arising from the forces on which the elasticity depends. The last part of the expression is essentially negative, or at least cannot be